Question

tan^2A - sin^2A = tan^2A sin^2A​

Answer 1

To prove:

• tan²A-sin²A=tan²A.sin²A

Solution:

Consider LHS

$\sf\hookrightarrow tan^2A-sin^2A$

Apply formula:

• tan²A=sin²A/cos²A

$\hookrightarrow\sf \dfrac{sin^2 A }{cos^2 A}-sin^2A$

Taking LCM

$\hookrightarrow\sf \dfrac{sin^2 A-cos^2A. sin^2A }{cos^2A}$

Taking sin²A common

$\hookrightarrow\sf \dfrac{sin^2 A(1-cos^2A) }{cos^2A}$

Replace 1-cos²A with sin²A

$\hookrightarrow\sf \dfrac{sin^2 A(\bf{sin^2A}) }{cos^2A}$

Replace sin²A/cos²A with tan²A

$\hookrightarrow\sf sin^2A.tan^2A$

$\underline{\boxed{\cal {\blue{HENCE\; PROVED}}}}$

Learn trigonometric identities:

$\boxed{\begin{array}{l c }\textsf{ Important Trigonometric identities }:- \\ \\ \: \\\tt 1)\: sin^2\theta+ cos^2\theta=1 \\ \\ \tt 2)\:sin^2\theta= 1-cos^2\theta \\ \\\tt 3)\:cos^2\theta=1-sin^2\theta \\ \\ \tt 4)\:1+cot^2\theta=\tt{cosec}^2 \, \theta \\ \\\tt5)\: \tt{cosec}^2 \, \theta-cot^2\theta =1 \\ \\ \tt6)\:\tt{cosec}^2 \, \theta= 1+cot^2\theta \\\ \\\tt 7)\:sec^2\theta=1+tan^2\theta \\ \\ \tt8)\:sec^2\theta-tan^2\theta=1 \\ \\ \tt 9)\:tan^2\theta=sec^2\theta-1 \end{array}}$

Answer 2

Answer:

To prove:

tan²A-sin²A=tan²A.sin²A

Solution:

Consider LHS

\sf\hookrightarrow tan^2A-sin^2A↪tan

2

A−sin

2

A

Apply formula:

tan²A=sin²A/cos²A

\hookrightarrow\sf \dfrac{sin^2 A }{cos^2 A}-sin^2A↪

cos

2

A

sin

2

A

−sin

2

A

Taking LCM

\hookrightarrow\sf \dfrac{sin^2 A-cos^2A. sin^2A }{cos^2A}↪

cos

2

A

sin

2

A−cos

2

A.sin

2

A

Taking sin²A common

\hookrightarrow\sf \dfrac{sin^2 A(1-cos^2A) }{cos^2A}↪

cos

2

A

sin

2

A(1−cos

2

A)

Replace 1-cos²A with sin²A

\hookrightarrow\sf \dfrac{sin^2 A(\bf{sin^2A}) }{cos^2A}↪

cos

2

A

sin

2

A(sin

2

A)

Replace sin²A/cos²A with tan²A

\hookrightarrow\sf sin^2A.tan^2A↪sin

2

A.tan

2

A

\underline{\boxed{\cal {\blue{HENCE\; PROVED}}}}

HENCEPROVED

Learn trigonometric identities:

\begin{gathered}\boxed{\begin{array}{l c }\textsf{ Important Trigonometric identities }:- \\ \\ $\: \\\tt 1)\: sin^2\theta+ cos^2\theta=1 \\ \\ \tt 2)\:sin^2\theta= 1-cos^2\theta \\ \\\tt 3)\:cos^2\theta=1-sin^2\theta \\ \\ \tt 4)\:1+cot^2\theta=\tt{cosec}^2 \, \theta \\ \\\tt5)\: \tt{cosec}^2 \, \theta-cot^2\theta =1 \\ \\ \tt6)\:\tt{cosec}^2 \, \theta= 1+cot^2\theta \\\ \\\tt 7)\:sec^2\theta=1+tan^2\theta \\ \\ \tt8)\:sec^2\theta-tan^2\theta=1 \\ \\ \tt 9)\:tan^2\theta=sec^2\theta-1$\end{array}}\end{gathered}